Prove that the feet of the perpendiculars drawn from the vertices of a parallelogram onto its diagonals are the vertices of a parallelogram.

Using AAS congruence between the right triangles formed at each foot of perpendicular, $$\triangle DPO \cong \triangle BRO$$ gives $$PO=RO$$, and similarly $$\triangle AOS \cong \triangle COQ$$ gives $$OS=OQ$$. Since the diagonals PR and QS of quadrilateral PQRS bisect each other, PQRS must be a parallelogram.